Chemistry Reference
In-Depth Information
X
N
r
k
1
a
k
.1
C
r
k
/
;
g.0/
D
ˇ
S
k
D
1
and if we assume that
X
N
r
k
1
a
k
.1
C
r
k
/
1;
(4.46)
kD1
then g.0/
ˇ
S
, so there is a root between each pair of consecutive poles and there
is either a positive and a negative root, or zero and a positive or negative root, or
zero with multiplicity two before the smallest pole. So all roots are real and they are
between
1 and +1 if g.
1/<ˇ
S
C
1.
Example 4.10.
As a special case consider linear price and cost functions i.e., p
D
f.Q/
D
A
BQ and C
k
.x
k
/
D
d
k
C
c
k
x
k
respectively. Then from Example 1.1
we know that r
k
D
1=2 for all k. In this case
a
k
C
ˇ
S
2
.a
k
C
ˇ
S
/.1
C
r
k
/
D
<1;
if a
k
C
ˇ
S
is below 2, which always holds if both a
k
and ˇ
S
are less than or equal
to 1 and at least one of them is below one. Condition (4.46) also has the special
form
X
N
1
2
a
k
1;
k
D
1
which clearly holds if N
2 and 0
a
k
1. In this special case
N
N
X
X
r
k
.
a
k
2ˇ
S
/
2
a
k
.1
C
r
k
/
D
a
k
C
2ˇ
S
4
a
k
g.
1/
D
;
k
D
1
k
D
1
so the equilibrium is locally asymptotically stable if
X
N
a
k
C
2ˇ
S
4
a
k
<ˇ
S
C
1:
kD1
Notice that this relation can be rewritten as
N
4
a
k
1
!
<1
X
X
N
2
a
k
4
a
k
:
ˇ
S
kD1
kD1
If the firms select identical adjustment schemes, then a
1
D D
a
N
D
a,sothis
relation simplifies to
ˇ
S
2N
4
a
1
<1
Na
4
a
;
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